Àmbit municipal

We have seen in Section 5 in the main text that some results are supportive of the static theory, confirmed by a formal regression analysis in Section 6. Yet it is also well-established in experimental work that models with Bayesian rationality and risk-neutral agents may not provide the best fit for the data.

The general assumption of our model is that people are risk-neutral. As a first check we will see if this assumption is warranted. Next, we will analyze if loss-aversion may play a role in people’s behavior. We present the results for “standard parameters” but emphasize that we have also tried other specifications without being able to improve the fit. Finally we will check if various forms of alternative information updating provide a better fit with the

data. These approaches usually depend on some parameter(s). Our approach is to vary this parameter and see how the variation improves the overall fit of the alternative model to the data. In this appendix we focus on the static decision only.

A.1

Risk and Loss Aversion

Risk Aversion. One persistent finding from the Section VII is that traders exhibit a general tendency to act as contrarians. One might thus entertain the idea that traders act as contrarians because of risk-aversion. We can go about examining this by computing the optimal action when people have a concave utility function. We checked this employing both CARA and CRRA utility functions:

utilityCARA(payoff|action) =−e

ρ·payoff

, utilityCRRA(payoff|action) =

payoff1−γ

1−γ .

Theoretically, the CARA utility function is the superior choice in the framework since we can ignore income effects.

For each type we determined the optimal action given the respective utility function and compared it to the action taken by the subjects. Within a setup with risk-aversion, a pass is indeed an action that has payoff consequences and may be optimal for some posterior probabilities. Usually, as prices (and thus the probability of a high outcome) rise, the optimal action changes from a buy to a pass to a sell. Risk-aversion biases decisions against buys and holds, because sells yield an immediate cash flow, whereas holding the stock exposes the subject to the risky future payoff. The larger the risk-aversion coefficient, the stronger the bias against buying.

Computing the expected utilities we find, however, that the performance of a model with risk aversion is worse for all reasonable levels of risk aversion. For CRRA with log- utility (γ = 1), it is 67% , which is below the risk-neutral model (70%) and the fit is only 42% for the S2 types; for CARA with ρ = 2 it is 51% (the fit rises asρ declines). Asρ declines,

we capture more of the behavior by S3 types but less of the behavior by S2 types. Note

that as ρ decreases, we move closer to risk neutrality. Table I. contains the details of these specifications.

Overall, we conclude that the assumption of risk-neutrality captures behavior quite well, with risk-aversion playing at most a negligible role.

Loss-Aversion — S-Shaped Valuation Functions. A host of experimental work in prospect theory following Kahneman and Tversky (1979) has indicated that people pick choices based on change in their wealth rather than on levels of utilities. These costs and

benefits of changes in wealth are usually assessed with valuation functions that are S-shaped. Kahnemann and Tversky suggested the following functional form

V(∆wealth|action) = ( (∆wealth)α for ∆wealth≥0 −γ(−∆wealth)β for ∆wealth<0

where ∆wealth is the change in wealth and α, β, γ are parameters. A common specification for the parameters stemming from experimental observations is α =β = 0.8 and γ = 2.25 (Tversky and Kahneman (1992)).

As with risk aversion, the performance of this model applied to our setup is much worse than the performance of the rational model. For parameters as estimated by Tversky and Kahneman (1992), the fit is below 49%. Table II. illustrates this observation for the above parameters as well as for one other configuration.1

A.2

Decision Rule: Prior Actions or No Updating

One alternative decision rule formulation is that of na¨ıve traders who ignore the history and who simply stick to their prior action. As such,S1 types always sell,S3 types always buy and

S2 types pick the action that is prescribed at the initial history. For instance, with negative

U-shape, S2 traders always sell.

This specification does no better than the rational model, fitting 71% of the data; broken up by type the fit is similar to the rational model. Moreover, with this alternative model, we cannot accommodate passes as ‘weak buys’ because this would be contrary to the spirit of ‘no changes of the action’. Indeed this illustrates the first weakness: a model based on people choosing their prior action will not help us to understand any changes in behavior that might have occurred, in particular not for S1 and S3 types. Since the econometric analysis

has already revealed that traders are sensitive to the price, this decision rule is rather weak. A weaker variation of the ‘stick to the prior action’-theme has traders ignore the his- tory altogether but remain mindful of the price. Traders thus act based only their prior expectation: if the price exceeds it, they sell, if the price is below it, they buy.

And indeed about 75% of people take an action that is in accordance with their prior expectation. For instance, for the S3 types this means that they do not buy when they

should be buying, or for the S2 types that they do not herd when they should be herding.

Table III. contains the details of the fit that is obtained under the two specifications outlined here.

1

Arguably, we are only using one part of the tools developed in prospect theory, S-shaped valuations, and ignore that other component, decision weights. However, the latter relate to re-scaled probabilities which we analyze separately so as to be able to distinguish the effects of the two components.

A.3

Probability Scaling and Shifting

A yet weaker version of the no-updating alternative rule is probability shifting, whereby traders underplay (overplay) low (high) probabilities coming from the observed historyHt₋1.

Alternatively, traders may overstate the probabilities of their prior expectations; we present results from the latter but point out, that the former yields similar insights. The usual sym- metric treatment of this under- or overstating of probabilities is to transform probability p

into f(p) as follows2

f(p) = p

α pα_{+ (1−p)}α.

Parameter values α > 1 are associated with S-shaped re-valuations (high probabilities get overstated, low probabilities understated), α < 1 with reverse S-shaped valuations (high probabilities get understated, low probabilities overstated). Note that transformation f(p) applied to probabilities of all three states do not yield a probability distribution. However, when employed properly in the conditional posterior expectation the transformation achieves the effect of a probability distribution.

Consequently, when modeling an overconfident trader who puts more weight on his prior signal we would apply anα >1 re-scaling on the initial probabilities. Alternatively, one can also model slow updating directly by applying an α <1 re-scaling to the posterior probabil- ities. Of course the effect will be similar: in both cases the histories or updated probabilities would be less important to traders than under the rational model. We considered both specifications.

Here we report the results where Pr(V|H1)×Pr(S|V) has been re-scaled with anα >1;

downward scaled probabilities of the history Pr(V|Ht) yield similar insights.

Comparing the results listed in Table IV. with those in Table I. in the main text, one can see that the fit of prior overweighing hardly improves for theS1 andS3types. Moreover, while

the total fit does improve relative to the rational model, it does not improve dramatically. Most of the improvement stems from contrarian trades that are now given a rationale. At the same time, re-scaling does a poor job explaining herd-behavior of any sort.

2

There are various other forms for these switches, e.g. non-symmetric switches where the effects are stronger (or weaker) for larger probabilities. The interpretation and implementation of such asymmetric shifts does, however, become difficult if not impossible with three states. Of the various possible specifications we only pick a few as the spirit of all re-scalings is similar: updating is slowed.

Inf_{, one re-scales} pα _{by itself and the counter-probability; alternatively, ifp}

i signifies the probability of

A.4

Error Correction Provisions

Inspired by level-k reasoning (see Costa-Gomes, Crawford and Broseta (2001)) and Quantal Response Equilibria (see McKelvey and Palfrey (1995) and McKelvey and Palfrey (1998)), we will contemplate an alternative specification for hampered updating in which agents do not trust that their peers act fully rationally. In the rational model, consider a buy without herding in state Vi: this event occurs with probability βi =.25/2 +.75·Pr(S3|Vi) (recalling

that.25/2 is the probability of a noise buy). Now imagine that instead subjects believe that only fraction δ of the informed buyers act rationally and that the remaining 1−δ take a decision at random. Then the probability of a buy in state Vi becomes

βi =.25 +.75((1−δ)/2 +δ·Pr(S3|Vi)).

The task is then to find the δ for which this specification yields the best fit with the data. We obtained the best fit for δ = 2/15. However, compared to the rational model the improvement of the fit is minor (see Table V.): the rational fit is 70% vs. 73% with error correction provisions.

An alternative interpretation for this error correction is that the level of noise trading is perceived higher than it actually is because other subjects act randomly: if δ = 2/15, then this translates into a factual noise level of 90%. As the informational impact of each transaction on the subject’s beliefs is dampened, after any history the private signal has a larger impact than under the rational model. This specification is thus in spirit similar to probability shifting, but focuses on the idea that subjects believe that others either ignore their signals or are simply unable to interpret it correctly.

A variation on this error correction theme is a specification in which a subject believes that fraction 1−δ act randomly but the subject assumes that the remaining fractionδ takes this irrationality into account and reacts rationally to it. The difference to the first specification is that in the first, the subject not only assumes irrationality on the part of informed traders but also considers himself to be the only informed trader to take this into consideration. Now we instead allow a later subject to believe that his predecessors are also aware of the possible irrationality on the part of informed traders and employ this knowledge in their decision-making. Consequently, in the first specification, S3 traders would never have been

presumed to rationally sell, whereas in the second specification such behavior is admitted as rational.3

Alas, as with the simple error correction, we do not obtain a substantially better

3

Rather than directly implementing level-k reasoning or Quantal Response Equilibria, we choose our alternative specification because it is an unusually complex task for the subjects to calculate these more general measures of naive reasoning with 4 different known types of traders (noise traders and three types of informed trader). Moreover, there is a subtle difference of our approach to the way that Quantal Response

fit with the data, as can be gleaned from Table V.: we obtained the best fit for δ = 0 in which case people act only on the basis of their prior expectation and do not update. For

δ=.22 (presented in the table; the figures forδ = 0 coincide with those of the no-updating case), the fit is best for treatments 1-3 (treatments 4-6 have the best fit for δ = 0). In the latter case, the improvement for treatments 1-3 only is from 69.8% to 76.1%.

In summary, a model specification in which agents recursively take their predecessor’s decisions as prone to error provides a worse fit with a data than the overweighing of one’s own signal. Compared to the rational model there is an improvement of fit, though it is small.

A.5

Summary of Alternative Behavioral Explanations

While forms of slow updating improve the fit of the data slightly, no alternative model is capable of providing a convincing explanation for the results. Slow updating, overweighing of one’s own signal, and overestimating noise trading are essentially very similar, and also have strong similarities to a strategy of following the prior (which is a policy of zero updating).

Several studies (Drehmann, Oechssler and Roider (2005) and Cipriani and Guarino (2005)) have already identified that when prices rise, people with high signals tend to act as contrarians, i.e. they sell. There are multiple possible explanations, ranging from risk aversion (which we refute) to slow or no updating. We observe the same kind of end-point behavior by theS3 types. Symmetrically, theS1 types should exhibit similar behavior when

prices approach the lower bound. However our data rarely involves prices that fall to a suf- ficient extent to examine the symmetric claim, since in general across all treatments, prices tend to tentatively rise. Note that the end-point effect should also influence the S2 types,

because whatever mechanism or cognitive bias leads S3 types to sell for high prices should

apply in the same manner to S2 types.

Irrespective of which hypothesis is correct, if the end result is observationally equivalent to slow updating then this has a profound effect on how much herding or contrarian behavior one might expect to see: when people update slowly, it takes longer for them to reach a

Models can be implemented in models with and without prices. In an informational cascade without prices a deviation from the cascading action is, in principle, a deviation from rationality. With moving prices, such a simple observation can no longer be made, neither is it possible for subjects to determine if there is a genuine error. Our notion of overweighing noise is therefore a simple means for subjects to model the lack of trust in predecessors’ actions, without implying a definitive or systematic direction of the error. Traders thus act as if the proportion of noise traders were higher than 25% by downgrading the quality of information extracted

from the history of actions embodied in H_t−1 or qt. Finally, since we already have noise traders built into

the experiment, by opting to allow traders to increase their estimates of the percentage of expected noise trades above 25% our method is arguably an especially simple and intuitive rule of thumb which enables subjects to incorporate naive reasoning on the part of their peers. For more on rules of thumb by laboratory subjects in a herding context see Ivanov, Levin and Peck (2008).

(subjective) expectation for which they would herd. However, with slow updating, they will also be slower to reduce prices and thus it is conceivable that they herd when prices move “against” the herd.